.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/analytical/chromatography/plot_02_martin_synge_theoretical_plates.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_analytical_chromatography_plot_02_martin_synge_theoretical_plates.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_analytical_chromatography_plot_02_martin_synge_theoretical_plates.py:


Martin and Synge's theoretical plates from a peak's width
============================================================

Treating a column as `N` equilibration stages ("theoretical plates")
predicts a Gaussian peak whose width grows as :math:`t_R/\sqrt N`.
:func:`~chemistrykit.analytical.theoretical_plates` inverts that:
:math:`N=16(t_R/w_{base})^2=5.545(t_R/w_{1/2})^2`, and
:func:`~chemistrykit.analytical.plate_height` gives the plate height
:math:`H=L/N`. Here we measure the widths of simulated peaks and recover
the plate count they were built from.

.. GENERATED FROM PYTHON SOURCE LINES 15-41



.. image-sg:: /api/gallery/analytical/chromatography/images/sphx_glr_plot_02_martin_synge_theoretical_plates_001.png
   :alt: Same $t_R$, more plates $\Rightarrow$ narrower peak
   :srcset: /api/gallery/analytical/chromatography/images/sphx_glr_plot_02_martin_synge_theoretical_plates_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    N built in =    1000: from FWHM 0.7444 min -> N =    1001; from base width -> N =    1000; H = 249.8 um
    N built in =    5000: from FWHM 0.3328 min -> N =    5007; from base width -> N =    5000; H = 49.9 um
    N built in =   25000: from FWHM 0.1488 min -> N =   25044; from base width -> N =   25000; H = 10.0 um






|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.analytical import plate_height, simulate_chromatogram, theoretical_plates

    t = np.linspace(8.0, 12.0, 20001)
    column_length = 25.0  # cm
    fig, ax = plt.subplots(figsize=(7, 4.5))
    for N_true in (1000.0, 5000.0, 25000.0):
        peak = simulate_chromatogram(t, centers=[10.0], N=N_true)
        above = t[peak >= peak.max() / 2]
        w_half = above.max() - above.min()
        sigma = 10.0 / np.sqrt(N_true)
        N_half = theoretical_plates(10.0, w_half, width_type="half_height")
        N_base = theoretical_plates(10.0, 4.0 * sigma, width_type="base")
        H = plate_height(column_length, N_half)
        print(f"N built in = {N_true:7.0f}: from FWHM {w_half:.4f} min -> N = {N_half:7.0f}; from base width -> N = {N_base:7.0f}; H = {H * 1e4:.1f} um")
        ax.plot(t, peak / peak.max(), label=f"N = {N_true:.0f} (H = {H * 1e4:.0f} $\\mu$m)")
        ax.hlines(0.5, 10.0 - w_half / 2, 10.0 + w_half / 2, color=ax.lines[-1].get_color(), linestyle=":")

    ax.set_xlabel("time (min)")
    ax.set_ylabel("normalized signal")
    ax.set_title(r"Same $t_R$, more plates $\Rightarrow$ narrower peak")
    ax.legend()
    plt.tight_layout()
    plt.show()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.061 seconds)


.. _sphx_glr_download_api_gallery_analytical_chromatography_plot_02_martin_synge_theoretical_plates.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_martin_synge_theoretical_plates.ipynb <plot_02_martin_synge_theoretical_plates.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_martin_synge_theoretical_plates.py <plot_02_martin_synge_theoretical_plates.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_martin_synge_theoretical_plates.zip <plot_02_martin_synge_theoretical_plates.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_